OOn the D(–1)/D7-brane systems
M. Bill`o a , M. Frau a , F. Fucito b , L. Gallot c ,A. Lerda d and J.F. Morales b a Universit`a di Torino, Dipartimento di Fisica,and I.N.F.N. - sezione di Torino,Via P. Giuria 1, I-10125 Torino, Italy b I.N.F.N. - sezione di Roma Tor Vergataand Universit`a di Roma Tor Vergata, Dipartimento di FisicaVia della Ricerca Scientifica, I-00133 Roma, Italy, c Laboratoire d’Annecy-le-vieux de Physique Th´eorique LAPThUniv. Grenoble Alpes, Univ. Savoie Mont-Blanc and CNRS,F-74000, Annecy, France d Universit`a del Piemonte Orientale,Dipartimento di Scienze e Innovazione TecnologicaViale T. Michel 11, I-15121 Alessandria, Italyand I.N.F.N. - sezione di Torino,Via P. Giuria 1, I-10125 Torino, Italy
E-mail: billo,frau,[email protected]; fucito,[email protected];[email protected]
Abstract
We study non-perturbative effects in supersymmetric U( N ) gauge theories in eight dimen-sions realized by means of D(–1)/D7-brane systems with non-trivial world-volume fluxes turnedon. Using an explicit string construction in terms of vertex operators, we derive the action forthe open strings ending on the D(–1)-branes and exhibit its BRST structure. The space ofvacua for these open strings is shown to be in correspondence with the moduli space of gen-eralized ADHM gauge connections which trigger the non-perturbative corrections in the eight-dimensional theory. These corrections are computed via localization and turn out to dependon the curved background used to localize the integrals on the instanton moduli space, andvanish in flat space. Finally, we show that for specific choices of the background the instantonpartition functions reduce to weighted sums of the solid partitions of the integers. Keywords: N = 2 SYM theories, instantons, D-branes a r X i v : . [ h e p - t h ] J a n ontents (cid:48) and D7 (cid:48) /D7 strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 The no-force condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 (cid:48) and D7 (cid:48) /D(–1) strings . . . . . . . . . . . . . . . . . . . . . . . . . 133.4 The no-force condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 ε -background . . . . . . . . . . . . 194.3 The BRST structure of the moduli spectrum . . . . . . . . . . . . . . . . . . . . . 22 k = 1 case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.2 Explicit results at higher instanton numbers . . . . . . . . . . . . . . . . . . . . . . 27 Non-perturbative effects in gauge theories can be rephrased in the language of string theoryby considering the effects induced by branes of lower dimensions distributed along the world-volume of branes of higher dimensions. The prime example of this construction is representedby the D(–1)/D3-brane system which realizes the ADHM moduli space of gauge instantons infour-dimensional gauge theories [1–7]. The massless sector of the open strings with at least oneend-point on the D(–1)-branes accounts for the moduli of the gauge instanton solutions in fourdimensions, and their effective action can be recovered by computing string scattering amplitudeson disk diagrams connecting the two stacks of branes [3–6].This analysis can be generalized to other brane systems. In particular, new stringy non-perturbative corrections to four-dimensional gauge theories can be obtained by adding EuclideanD3-branes wrapping non-trivial cycles of the internal space. When orientifold planes are inserted,1hese exotic instanton configurations [8–15] can generate non-perturbative effects in the effectiveaction which are prohibited in perturbation theory, like for instance certain Majorana mass termsor Yukawa couplings, which may be relevant for phenomenological applications.These methods can also be used to study non-perturbative effects in eight-dimensional fieldtheories by considering systems of D7-branes in presence of D-instantons. Indeed, the prepotentialgenerated by exotic instantons in O( N ) gauge theories in eight dimensions has been studied in[16, 17] by means of a system made of D(–1) and D7-branes on top of an orientifold O7-plane,allowing to explicitly test the non-perturbative heterotic/Type I string duality [18]. More generalD p /D p (cid:48) systems can be also considered after turning on fluxes for the Neveu-Schwarz B -field, insuch a way that supersymmetry is restored in the vacuum [1, 19].The peculiarity of all these exotic or higher dimensional instanton systems is that they lack thebosonic moduli connected to the size and the gauge orientation of the instanton configurations.Moreover, they possess extra fermionic zero-modes, in addition to those connected with the brokensupersymmetries, which can make their contributions to the effective action vanish if they are notproperly lifted or removed. The problems related to the presence of these extra fermionic zero-modes can be cured in various ways, for example by adding orientifold or orbifold projections,or more generically by considering curved backgrounds. However, in all these cases the non-perturbative configurations always remain point-like since no bosonic moduli can account for anon-zero size. The fundamental reason for this is that in all these exotic configurations there areeight directions with mixed Dirichlet/Neumann boundary conditions, differently from the standardcases in which there are only four mixed directions.The non-perturbative contributions to the gauge theory effective action are obtained after anexplicit evaluation of the instanton partition functions which are expressed as integrals over theinstanton moduli space. This is possible thanks to the localization techniques introduced in [20] forthe N = 2 super Yang-Mills theories in four dimensions, building also on previous results in [21].In the end, the instanton partition functions are reduced to integrals of rational functions whichcan be performed using standard complex analysis methods. Actually, the poles of these rationalfunctions that contribute to the integrals can be put in one-to-one correspondence with sets ofYoung tableaux which in turn are related to the partitions of the integer numbers [20, 22–26].When this approach is used for gauge theories in dimensions higher than four, higher dimensionalYoung tableaux appear and a connection with the planar and solid partitions of the integersarises [24, 27]. The instanton partition functions are thus expressed as weighted sums over integer,plane or solid partitions. Therefore, studying these systems may be interesting not only for thestring theory applications we mentioned above, but also for mathematical reasons, related directlyto such weighted sums (the generating function of solid partitions is not yet known in closed form)but also, for instance, in the study of Donaldson-Thomas invariants of Calabi-Yau four-folds [28].Recently, the instanton partition functions of a pure supersymmetric U( N ) gauge theory in ninedimensions and of its conformal extension with a U( N ) flavor symmetry have been analyzed fromthis point of view in [27,29] and shown to be given by a simple all-instanton plethystic exponentialformula. More recently, these systems have been generalized to non-conformal set-ups with aU( M ) flavor symmetry in [30] where also the instanton contributions to the chiral correlators havebeen computed. In all cases it turns out that the non-perturbative sectors are described by setsof moduli that share some features of both the standard and exotic instantons. Thus, in [27, 29] ithas been conjectured that they might arise from the open strings of a D0/D8-brane system with2 Neveu-Schwarz B -field and with anti-D8 branes in the background.The aim of this paper is to provide an explicit string theory derivation of the moduli space andof the instanton partition functions studied in [27, 29]. We consider a D(–1)/D7-brane system inType II B string theory and do not introduce any anti-branes which would make the configurationunstable, but instead turn on a magnetic flux along the world-volume of the D7-branes . Moreprecisely, we start from a stack of ( N + M ) D7-branes to describe a gauge theory in eight dimensions;then we introduce a constant magnetic field on the first N D7-branes to break the gauge groupto U( N ) × U( M ), maintaining stability and supersymmetry. Finally, we add k D(–1)-branes. Theopen strings with at least one end-point on the D-instantons describe the moduli space of thenon-perturbative configurations. The spectrum of the open strings that start and end on the D(–1)-branes is standard and includes, among others, the bosonic moduli describing the positions ofthe instantonic branes in the world-volume of the seven-branes. The spectrum of the mixed stringsstretching between the D(1) and the D7-branes is instead peculiar because, despite the presenceof eight directions with mixed Dirichlet/Neumann boundary conditions, due to the presence ofthe magnetic flux, it comprises a set of bosonic moduli that can be associated to the size andorientations of the instanton configurations in eight dimensions.We also provide a detailed analysis of the vertex operators associated to all moduli and usethem to derive the effective instanton action from disk amplitudes. After introducing vacuumexpectation values for the scalar fields on the D7-branes and turning on a closed-string backgroundwith Ramond-Ramond fluxes, which is known to mimic the so-called Ω-background in the modulispace [31], we compute the instanton partition function using localization. Our results agree withthose in [27, 29]; in particular for a suitable choice of the background the instanton partitionfunction reduces to a weighted sum over the solid partitions of the integers. Our derivation alsosuggests a possible consistent generalization of the results of [27, 29] in which the Ω background isless constrained; we will investigate this possibility in a future work.Finally, we show that the moduli space of vacua of the matrix theory defined on the D(–1)-branes is compatible with an ADHM construction of instanton connections in eight dimensions,which has also been recently discussed in [32]. In the concluding section we comment on thesignificance of our results for the eight-dimensional gauge theory, which receives non-perturbativecorrections from these instanton configurations only in its U(1) part and only in curved space.Our notations and conventions, together with some more technical material, are collected in theappendices.
In Type II B string theory, we consider a stack of ( N + M ) D7-branes aligned along the directions µ, ν, . . . = 1 , . . . , N + M ) and sixteen supercharges. The part of the effective action which only depends on thegauge field strength can be written as S D7 = S + S + · · · . (2.1) Notice that this magnetic flux is an open string background and thus is not represented by a Neveu-Schwarz B -field which belongs to the closed string sector. − − − − − − − − ∗ ∗ Table 1:
The D7-branes are aligned along the first eight directions.
Here S is the quadratic Yang-Mills action in eight dimensions S = 12 g (cid:90) d x tr (cid:0) F (cid:1) (2.2)with a dimensionful gauge coupling constant g ≡ πg s (2 π √ α (cid:48) ) , (2.3)( g s is the string coupling and √ α (cid:48) is the string length), while S is a quartic action of the form S = − λ (cid:90) d x tr (cid:0) t F (cid:1) − i ϑ π ) (cid:90) d x tr (cid:0) F ∧ F ∧ F ∧ F (cid:1) (2.4)where λ ≡ π g s (2.5)is a dimensionless coupling, and ϑ is the vacuum angle, which in string theory is identified withthe scalar field C of the RR sector according to ϑ = 2 π C . (2.6)The eight-index tensor t appearing in the first term of (2.4) is such that [33] tr (cid:0) t F (cid:1) ≡ t µ µ ··· µ µ tr (cid:0) F µ µ · · · F µ µ (cid:1) (2.7)= tr (cid:16) F µν F νρ F λµ F ρλ + 12 F µν F ρν F ρλ F µλ − F µν F µν F ρλ F ρλ − F µν F ρλ F µν F ρλ (cid:17) . Finally, the ellipses in (2.1) stand for α (cid:48) corrections containing at least five field strengths or theircovariant derivatives.The gauge field strength F µν is actually part of a scalar superfield Φ( x, θ ) in eight dimensions,defined as Φ( x, θ ) = φ ( x ) + √ θ Λ( x ) + 12 θσ µν θ F µν ( x ) + . . . (2.8)where θ is the fermionic superspace coordinate, φ is a complex scalar and Λ is an eight-dimensionalchiral fermion . In terms of the superfield Φ, the quartic action (2.4) (plus its supersymmetriccompletion) can be written as S = τ (cid:90) d x d θ tr (cid:0) Φ (cid:1) + c.c. (2.9) For an explicit definition of t , see for instance Appendix B of [34]. See Appendix A for our conventions on spinors and Dirac matrices. τ is the complexified string coupling τ = C + i g s . (2.10)In principle the quartic action (2.9) can receive quantum corrections and takes the general form S (cid:48) = (cid:90) d x d θ F (Φ , τ ) + c . c . (2.11)where F (Φ , τ ) is the (holomorphic) prepotential.We now introduce a (constant) background field on the first N D7-branes. The other M D7-branes, which remain without background field, will be called from now on D7 (cid:48) -branes. In this waythe initial gauge symmetry group U( N + M ) is broken to U( N ) × U( M ). In particular, we considera background that corresponds to a constant flux on the D7-brane world-volume described by2 πα (cid:48) F (0) = f − f f − f f − f f − f N × N . (2.12)In the following we will specify the four parameters f , . . . , f in such a way to preserve somesupersymmetry, but for the time being we may consider them as arbitrary . The background(2.12) gives rise to four different types of open strings, as represented in Fig. 1: the 7/7 stringsstarting and ending on the N D7-branes with fluxes, the 7 (cid:48) /7 (cid:48) strings starting and ending on the M D7 (cid:48) -branes without fluxes, and the 7/7 (cid:48) or 7 (cid:48) /7 strings which start and end on branes of differenttype. The 7/7 and 7 (cid:48) /7 (cid:48) strings are untwisted and contain exactly the same physical states of theoriginal strings. In particular they give rise to the same effective action (2.1) for the group U( N )(in presence of the background (2.12)) and for the group U( M ), respectively. The 7/7 (cid:48) or 7 (cid:48) /7strings, instead, are twisted and their spectrum is completely different, depending on the valuesof the fluxes. (cid:48) and D7 (cid:48) /D7 strings To analyze the spectrum of the D7/D7 (cid:48) and D7 (cid:48) /D7 open strings, it is convenient to group thespace-time coordinates in pairs and introduce the complex combinations z I = x I − + i x I √ z I = x I − − i x I √ I = 1 , , , ,
5. The directions with I = 1 , , , I = 5 is transverse. In this complex notation, the background (2.12) implies that in the 7/7 (cid:48) We notice that the flux breaks Lorentz invariance in the eight-dimensional space, so that the preserved super-symmetry can be also viewed as a two-dimensional supersymmetry in the transverse space. . . . . .N D7-branes M D7 (cid:48) -branes Figure 1:
A background flux is turned on on the N D7 branes. The D7 / D7 and D7 (cid:48) /D7 (cid:48) open strings areuntwisted, the mixed ones (depicted as dashed) are instead twisted. sector, the complex string coordinates (cid:8) Z I ( z ) , Z I ( z ) (cid:9) and their fermionic partners (cid:8) Ψ I ( z ) , Ψ I ( z ) (cid:9) with I = 1 , , , θ I given by e π i θ I = 1 − i f I f I . (2.14)This means that they have the following monodromy properties around the origin of the world-sheet: Z I (e π i z ) = e π i θ I Z I ( z ) , Z I (e π i z ) = e − π i θ I Z I ( z ) , (2.15)and Ψ I (e π i z ) = ± e π i θ I Ψ I ( z ) , Ψ I (e π i z ) = ± e − π i θ I Ψ I ( z ) . (2.16)Here z is a point in the upper half complex plane and the sign in (2.16) is + for the Neveu-Schwarz (NS) boundary conditions and − for the Ramond (R) ones. In [35], the bosonic andfermionic conformal field theory for generic twist parameters has been worked out in detail andwe summarize the most relevant features in Appendix B. Here we just recall that the monodromyproperties (2.15) of the bosonic coordinates do not allow for the existence of zero-modes associatedto momentum in these directions.The bosonic and fermionic coordinates (cid:8) Z ( z ) , Z ( z ) (cid:9) and (cid:8) Ψ ( z ) , Ψ ( z ) (cid:9) are instead un-twisted. They have the standard mode expansion of untwisted fields, but with Dirichlet/Dirichletboundary conditions since they are transverse to both type of branes. Due to these boundaryconditions there are no bosonic zero-modes in these directions either. Thus, when all four θ I ’s aredifferent from zero, the momentum is not defined in any direction, and the corresponding stringexcitations represent non-dynamical degrees of freedom. In the 7 (cid:48) /7 sector we have to send f I to − f I .
6s mentioned above, we are interested in supersymmetric configurations. This means that thebackground (2.12) must be such that θ + θ + θ + θ ∈ Z . (2.17)Of course, if θ I = 0 for all I , the background vanishes and all sixteen supercharges are conservedon the branes. If two of the twists are zero, the system preserves eight supercharges while if onlyone of the θ I ’s vanishes the system has four conserved supercharges. If all twists are non-zeroand satisfy (2.17), the system preserves the minimal amount of supersymmetry, namely only twosupercharges are conserved. This is the case we will consider. For simplicity, we assume that0 < θ I < for all I ’s . With this choice, the supersymmetry condition (2.17) becomes θ + θ + θ + θ = 1 . (2.18)In this set-up, the condition that selects the physical states is (cid:98) N = 12 (cid:16) − (cid:88) I =1 θ I (cid:17) in the NS sector , (cid:98) N = 0 in the R sector . (2.19)where (cid:98) N is the sum of the bosonic and fermionic number operators (see (B.6) and (B.8)). Usingthe supersymmetry condition (2.18), we see that both the NS and R physical states must actuallyobey (cid:98) N = 0 . (2.20)Since the number operator is non-negative, this requirement can be satisfied only by the vacuum.In the NS sector, the physical vacuum | Ω (cid:105) NS is related to the SL(2 , R ) invariant Fock vacuum | (cid:105) by the action of four bosonic twist fields σ Iθ I ( z ) of conformal dimensions h ( σ Iθ I ) = 12 θ I (1 − θ I ) (2.21)and four fermionic twist fields s Iθ I ( z ) of conformal dimensions h ( s θ I ) = 12 θ I (2.22)as follows [36] | Ω (cid:105) NS = lim z → (cid:16) (cid:89) I =1 σ Iθ I ( z ) s Iθ I ( z ) (cid:17) e − ϕ ( z ) | (cid:105) . (2.23)Here ϕ ( z ) is the bosonic field that is used to describe the superghost system in the bosonizedformalism [37] and the coefficient − − The reason to exclude the value is merely to avoid the appearance of fermionic zero-modes in the NS sector. s Iθ I ( z ) can be written in the bosonized formalism as s Iθ I ( z ) = e i θ I ϕ I ( z ) (2.24)where ϕ I ( z ) is the field that bosonizes the fermionic string coordinates (cid:8) Ψ I ( z ) , Ψ I ( z ) (cid:9) accordingto [37] Ψ I ( z ) = e i ϕ I ( z ) , Ψ I ( z ) = e − i ϕ I ( z ) . (2.25)Of course, analogous bosonization formulas hold also for the untwisted coordinates (cid:8) Ψ ( z ) , Ψ ( z ) (cid:9) .Using these expressions, it is easy to realize that the sum of the charges of the five bosonic fields ϕ I appearing in the vertex operator corresponding to the NS vacuum minus the superghost chargeis an even integer, namely (cid:88) I =1 θ I + 0 − ( −
1) = 2 . (2.26)Denoting by F this combination of charges, we have( − F | Ω (cid:105) NS = + | Ω (cid:105) NS . (2.27)Let us now consider the R sector. In this case the physical vacuum is created out of the Fockvacuum by the same bosonic twist fields σ θ I ( z ) as before, but in the fermionic part we have thefollowing twist fields s Iθ I − ( z ) = e i ( θ I − ) ϕ I ( z ) (2.28)for I = 1 , . . . , S ± ( z ) = e ± i2 ϕ ( z ) (2.29)in the fifth complex untwisted direction. The presence of this spin field implies that the R vacuumis actually a two-component SO(2) spinor. Altogether, working in the standard ( − )-superghostpicture, in the R sector we find two physical vacuum states which in the bosonized formalism aregiven by | Ω , ±(cid:105) R = lim z → (cid:16) (cid:89) I =1 σ Iθ I ( z )e i ( θ I − ) ϕ I ( z ) (cid:17) S ± ( z ) e − ϕ ( z ) | (cid:105) . (2.30)The combination of charges that before we denoted by F takes the value 0 on | Ω , + (cid:105) R and thevalue − | Ω , −(cid:105) R . Thus we have( − F | Ω , ±(cid:105) R = ±| Ω , ±(cid:105) R , (2.31)showing that ( − F actually measures the chirality of R vacuum.If we define the GSO projector as P GSO = 1 + ( − F , (2.32)then, using (2.27) and (2.31), we see that it selects the vacuum in the NS sector and the componentwith positive chirality in the R sector, realizing in this way a supersymmetric physical spectrumcontaining just two states, the bosonic vacuum | Ω (cid:105) NS and the fermionic vacuum | Ω , + (cid:105) R . We stressonce more that these states do not carry any momentum and thus represent non-dynamical degrees8f freedom since they do not propagate in space. Moreover, they transform in the fundamental ofU( N ) and in the anti-fundamental of U( M ), since they arise from oriented open strings stretchingbetween the N D7-branes and the M D7 (cid:48) -branes.We conclude this analysis with a couple of comments. First, the GSO-projection in the NSappears to be different with respect to the one we are used to in the standard untwisted casewhen one removes the (tachyonic) vacuum and selects states with an odd number of fermionicoscillators. However, if one reformulates this standard GSO projection in terms of F , one canrealize that it is precisely given by (2.32). Secondly, the analysis we have described for the D7/D7 (cid:48) strings can be repeated for the D7 (cid:48) /D7 strings without any difficulty, and also in that case onefinds that the physical states are a non-dynamical scalar, corresponding to the NS vacuum, andthe chiral component of a non-dynamical fermion, corresponding the chiral component of the Rvacuum. Of course, these states of the 7 (cid:48) /7 sector transform in the anti-fundamental of U( N ) andin the fundamental of U( M ). We have seen that the condition (2.18) guarantees a supersymmetric physical spectrum. To com-plete our analysis we now show that the same condition also implies that the partition functionsof the mixed strings of the 7/7 (cid:48) and 7 (cid:48) /7 sectors vanish, thus leading to a no-force condition.Let us first consider the 7/7 (cid:48) open strings whose partition function is Z / (cid:48) = (cid:90) ∞ dt t Tr / (cid:48) (cid:16) P GSO e − πt L (cid:17) (2.33)where the GSO projector is defined in (2.32) and L is the total open string Hamiltonian consistingof the orbital part and of the ghost and superghost parts (see for instance [37] for details). Thetrace in (2.33) is computed over all excitations of the 7/7 (cid:48) strings with a + sign in the bosonicNS sector and a − sign in the fermionic R sector. This calculation is pretty standard (see forexample [34, 38] and references therein), but we would like to point out just a couple of subtleissues that arise in the present case. First of all, in the 7/7 (cid:48) sector, there are eight (real) twisteddirections and two (real) untwisted directions. This is precisely the case in which the odd spin-structure R( − F gives a non-vanishing contribution to the string partition function due to acancellation between the fermionic zero-modes in the two untwisted directions and the bosoniczero-modes of the superghost-sector [38–40]. Secondly, the presence of the background (2.12)along the D7-brane world-volume modifies the contribution of the zero-modes of the eight twistedbosonic coordinates by multiplying the eight-dimensional volume V by the factor (cid:81) I f I .Taking this into account and using the GSO projection (2.32), we find Z / (cid:48) = N M V (4 π α (cid:48) ) (cid:89) I =1 f I (cid:90) ∞ dt t (cid:34) (cid:89) I =1 ϑ ( θ I t ) ϑ ( θ I t ) + (cid:89) I =1 ϑ ( θ I t ) ϑ ( θ I t ) − (cid:89) I =1 ϑ ( θ I t ) ϑ ( θ I t ) − (cid:35) (2.34)where the factor N M accounts for the color multiplicity of each string excitation and the fourterms in the square brackets correspond to the contributions of the NS, NS( − F , R and R( − F spin-structures, respectively. Here, as usual, we have defined: ϑ ( z ) ≡ ϑ (cid:2) (cid:3)(cid:0) i z | i t ) , ϑ ( z ) ≡ ϑ (cid:2) (cid:3)(cid:0) i z | i t ) , ϑ ( z ) ≡ ϑ (cid:2) (cid:3)(cid:0) i z | i t ) , ϑ ( z ) ≡ ϑ (cid:2) (cid:3)(cid:0) i z | i t ) (2.35)9here ϑ (cid:2) ab (cid:3)(cid:0) i z | i t ) are the Jacobi ϑ -functions with characteristics (see for example [41]). They satisfythe Jacobi-Riemann identity (cid:89) I =1 ϑ ( z I ) − (cid:89) I =1 ϑ ( z I ) − (cid:89) I =1 ϑ ( z I ) − (cid:89) I =1 ϑ ( z I ) = − (cid:89) I =1 ϑ ( Z/ − z I ) (2.36)with Z = (cid:80) I z I . Applying this identity in our case where z I = θ I t and using the periodicityproperties of the Jacobi ϑ -functions, we can recast the previous identity in the following form (cid:89) I =1 ϑ ( θ I t ) + (cid:89) I =1 ϑ ( θ I t ) − (cid:89) I =1 ϑ ( θ I t ) − (cid:89) I =1 ϑ ( θ I t ) = 0 , (2.37)when the twist parameters satisfy the supersymmetry relation (2.18). Thus, inserting this relationinto (2.34), we find Z / (cid:48) = 0 . (2.38)In a completely similar way one can show that also for the other orientation the partition functionvanishes: Z (cid:48) / = 0 . (2.39)This fact, together with the vanishing of the partition function in the 7/7 and 7 (cid:48) / (cid:48) sectors dueto the BPS property of these branes, implies that our brane system is stable when the condition(2.18) is satisfied, and that one can pile up an arbitrary number of D7 and D7 (cid:48) -branes since thenet force in all channels is zero. To study some non-perturbative features in the effective theory defined on the world-volume ofthe D7 and D7 (cid:48) -branes, we add a stack of k D–instantons, i.e.
D-branes with Dirichlet boundaryconditions in all directions also known as D( − (cid:48) and7 (cid:48) /(–1) strings which connect the D-instantons and the D7 (cid:48) -branes. We now analyze the physicalstates in all these sectors. The open strings stretching between two D(–1)-branes are untwisted and have Dirichlet/Dirichletboundary conditions in all ten directions. Thus, they do not carry any momentum and describenon-propagating degrees of freedom. To describe the physical states in this sector, it is convenientto distinguish the eight directions, that are longitudinal to the D7 and D7 (cid:48) - branes from the twoones that are transverse, and organize the open string states in representations of Spin(8) × Spin(2).In the NS sector, the only physical states are those with one fermionic oscillator, correspondingto the ten-dimensional vector. Using the same complex notation introduced in the previous section,we can pack the eight (real) components of this vector along the D7 or D7 (cid:48) -branes into four complex10ariables which we denote by B I , together with their complex conjugates B I , with I = 1 , , , − V B I = B I √ I ( z ) e − ϕ ( z ) and V B I = B I √ I ( z ) e − ϕ ( z ) . (3.1)The remaining two components of the vector, along the transverse directions to the branes, giverise to the variables ξ and χ , whose vertex operators are V ξ = ξ Ψ ( z ) e − ϕ ( z ) and V χ = χ ( z ) e − ϕ ( z ) . (3.2)All these vertex operators are conformal fields of dimension 1 and possess an even F -charge, asone can see using the bosonization formulas (2.25); thus they are preserved by the GSO projection(2.32).In the R sector the only physical state is the vacuum. This state is actually a sixteen-componentspinor of Spin(10), which the GSO projection fixes to be anti-chiral. To describe these sixteencomponents we use the formalism of spin fields [37] and introduce the notation S ±±±± ( z ) = e ± i2 ϕ ( z ) e ± i2 ϕ ( z ) e ± i2 ϕ ( z ) e ± i2 ϕ ( z ) (3.3)to denote the spin fields in the first four complex directions, which have to be used together withthe spin fields S ± ( z ) in the fifth direction already introduced in (2.29). The physical ground stateof the R sector, in the ( − )-superghost picture, is therefore given bylim z → S ±±±± ( z ) S ± ( z ) e − ϕ ( z ) | (cid:105) (3.4)with a total odd number of − signs in the spin fields in order to have an even F -charge and bepreserved by the GSO projection. It is convenient to distinguish the cases in which the last spinfield is S + ( z ) or S − ( z ).If we have S + ( z ), the spin fields in the first four directions must have an odd number of minussigns. Clearly, there are eight possibilities. Using a notation that resembles the one used for thecomponents of the vector field in the NS sector, we denote these eight spin fields by S I ( z ) and S I ( z ) with I = 1 , . . . , M I and M I , respectively, and aredescribed by the following vertex operators of weight 1: V M I = M I S I ( z ) S + ( z ) e − ϕ ( z ) and V M I = S I ( z ) S + ( z ) e − ϕ ( z ) M I . (3.5)If instead we have S − ( z ), the spin fields in the first four directions must have an even numberof minus signs and there are again eight possibilities. We denote these spin fields by S ( z ), S ( z )and S IJ ( z ) = − S JI ( z ) with I, J = 1 , . . . , S IJ ( z ), but they are not independent since the following relation holds S IJ ( z ) = 12 (cid:15) IJKL S KL ( z ) . (3.6) The normalization factors in these and in all the following vertices are chosen for later convenience so that thefinal formulas look simpler. S − ( z ), whichwe denote by λ IJ , λ and η , are described by the following vertex operators V λ IJ = λ IJ √ S IJ ( z ) S − ( z ) e − ϕ ( z ) , (3.7)and V λ = λ S ( z ) − S ( z ) √ S − ( z ) e − ϕ ( z ) , V η = η S ( z ) + S ( z ) √ S − ( z ) e − ϕ ( z ) . (3.8)It is easy to verify that these vertex operators are conformal fields of weight 1. Furthermore, thefields λ and η are real, while the fields λ IJ are complex with their complex conjugates λ IJ givenby λ IJ = 12 (cid:15) IJKL λ KL , (3.9)as a consequence of the relation (3.6).Since there are k D-instantons, all variables appearing as polarizations in the above vertexoperators are ( k × k ) matrices transforming in the adjoint representation of U( k ). We now consider the open strings stretching between the D-instantons and the D7-branes. Inthis case there are two facts that one has to consider, namely that the first four complex spacedirections have mixed Dirichlet/Neumann boundary conditions and that there is a backgroundfield on the D7-branes given by (2.12). The combined net effect is that the first four complexdirections have an effective twist parameter given by (cid:0) − θ I (cid:1) .Taking this into account, one can see that the condition which selects the physical states inthe NS sector is (cid:98) N + 12 − (cid:88) I =1 θ I = 0 (3.10)where (cid:98) N is the total number operator. Inserting the constraint (2.18), we see that the physicalstates in the NS sector must have (cid:98) N = 0, which is possible only for the vacuum. Thus in thespectrum we have only one state whose polarization we denote by w . Its corresponding vertexoperator, in the ( − V w = w √ (cid:16) (cid:89) I =1 σ I − θ I ( z ) e i( − θ I ) ϕ I ( z ) (cid:17) e − ϕ ( z ) (3.11)which is a conformal field of weight 1 when the condition (2.18) is satisfied. The same condition alsoguarantees that the total F -charge is even so that this vertex is preserved by the GSO projection.In the R sector, the condition for physical states implies (cid:98) N = 0. Thus, also in this sectorthe vacuum is the only physical state. Due to the R boundary conditions in the untwisted fifthcomplex direction, the ground state is degenerate and is a two-component spinor denoted by µ ± .The corresponding vertex operator in the ( − )-superghost picture is V µ ± = µ ± (cid:16) (cid:89) I =1 σ I − θ I ( z ) e − i θ I ϕ I ( z ) (cid:17) S ± ( z ) e − ϕ ( z ) (3.12)12hich is a conformal field of weight 1. Exploiting the supersymmetry condition (2.18), it is easyto see that ( − F V µ ± = ± V µ ± (3.13)so that only µ + is selected by the GSO projection. To simplify the notation, we will denote µ + simply as µ .This analysis can be easily repeated for the strings with the opposite orientation stretchingbetween the D7-branes and the D-instantons. The result is that in the NS sector we have only oneGSO-even physical state w whose vertex operator is V w = w √ (cid:16) (cid:89) I =1 σ Iθ I − ( z ) e i( θ I − ) ϕ I ( z ) (cid:17) e − ϕ ( z ) , (3.14)while in the R sector the only GSO-even physical state is µ whose vertex operator is V µ = µ (cid:16) (cid:89) I =1 σ Iθ I − ( z ) e i θ I ϕ I ( z ) (cid:17) S + ( z ) e − ϕ ( z ) . (3.15)Taking into account the multiplicity of the D-instantons and of the D7-branes, the variables w and µ are ( k × N ) matrices transforming in the ( k , N ) representation of U( k ) × U( N ), while theconjugate variables w and µ are ( N × k ) matrices transforming in the ( k , N ) representation of thesame group. (cid:48) and D7 (cid:48) /D(–1) strings We now consider the strings connecting the D-instantons and the D7 (cid:48) -branes. The only butimportant difference with the previous case is that now there is no background field on the 7-branesso that the string fields in the first four complex directions have standard Dirichlet/Neumannboundary conditions without extra twists. Therefore, we can formally set θ I = 0 in the previousformulas. If we do this, we immediately realize that in the NS sector the physical states mustsatisfy the condition (cid:98) N + 12 = 0 , (3.16)which has no solution. Thus, there are no physical states in the NS part of the spectrum. Thisfact was already observed in [34] where the D( − µ (cid:48) ± . The corresponding vertexoperator is V µ (cid:48)± = µ (cid:48)± (cid:16) (cid:89) I =1 σ I ( z ) (cid:17) S ± ( z ) e − ϕ ( z ) (3.17)where σ I is the appropriate bosonic twist field when the I -th coordinate has Dirichlet/Neumannboundary conditions. This twist field has conformal dimension , and thus the vertex operator(3.17) has total weight 1. Furthermore, it is easy to check( − F V µ (cid:48)± = ∓ V µ (cid:48)± , (3.18)13o that only µ (cid:48)− , which we will simply call µ (cid:48) , survives the GSO projection.For the strings with opposite orientation that stretch between the D7 (cid:48) -branes and the D-instantons, we find a similar structure with no physical states in the NS sector, and only onephysical state in the R sector which we denote µ (cid:48) . Its vertex operator is V µ (cid:48) = µ (cid:48) (cid:16) (cid:89) I =1 σ I − ( z ) (cid:17) S − ( z ) e − ϕ ( z ) (3.19)where σ I − is the bosonic twist field corresponding to an I -th coordinate with Neumann/Dirichletboundary conditions, that is conjugate to the twist field σ I .Taking into account the multiplicity of the branes, we have that µ (cid:48) is a ( k × M ) matrixtransforming the ( k , M ) representation of U( k ) × U( M ), while µ (cid:48) is ( M × k ) matrix transformingthe ( k , M ) representation of this group. We now study the partition function of the open strings with at least one end-point on the D-instantons. In the ( − / ( −
1) sector, we simply have Z ( − / ( − = 0 (3.20)as a consequence of the BPS condition satisfied by the D(–1)-branes.Let us now consider the ( − / (cid:48) and the 7 (cid:48) / ( −
1) sectors. Recalling that the D7 (cid:48) -branes donot have any background field on their world-volume, the calculation of the partition functions Z ( − / (cid:48) and Z (cid:48) / ( − is exactly the same as the one described in detail in [34], which we brieflyrecall here. For the strings starting from the D-instantons we have Z ( − / (cid:48) = kM (cid:90) ∞ dt t (cid:20)(cid:16) ϑ (0) ϑ (0) (cid:17) + (cid:16) ϑ (0) ϑ (0) (cid:17) − (cid:16) ϑ (0) ϑ (0) (cid:17) − (cid:21) (3.21)where the prefactor accounts for the multiplicity of the branes, and the four terms inside thesquare brackets arise from the four spin structures with eight directions with Dirichlet/Neumannboundary conditions. We notice in particular that the second term, corresponding to the NS( − F sector, actually vanishes since ϑ (0) = 0, while the fourth term, corresponding to the R( − F spinstructure, is non-zero due to the cancellation of the fermionic zero-modes in the two directions withDirichlet/Dirichlet boundary conditions with the bosonic zero-modes of the superghost system [38].Exploiting the abstruse identity ϑ (0) − ϑ (0) − ϑ (0) = 0 , (3.22)we easily conclude that Z ( − / (cid:48) = − kM (cid:90) ∞ dt t . (3.23)The partition function for the other orientation corresponds to placing the D-instantons “on theother side” of the 7-brane, which is obtained with a parity transformation that reverses the signof the odd spin structure R( − F [38]. Then we have Z (cid:48) / ( − = kM (cid:90) ∞ dt t (cid:20)(cid:16) ϑ (0) ϑ (0) (cid:17) + (cid:16) ϑ (0) ϑ (0) (cid:17) − (cid:16) ϑ (0) ϑ (0) (cid:17) + 1 (cid:21) . (3.24)14he abstruse identity (3.22) now implies that Z (cid:48) / ( − = 0 . (3.25)Let us now consider the contribution of the open strings connecting the D-instantons to theD7-branes. In this case, due to the presence of the background (2.12) on the 7-branes, we havean extra twist to take into account with respect to the previous cases. In particular, the partitionfunction Z ( − / is obtained from (3.21) by twisting the ϑ -functions by − θ I t in each complexdirection and replacing M by N , which is the number of D7-branes. In other words we have Z ( − / = kN (cid:90) ∞ dt t (cid:34) (cid:89) I =1 ϑ ( − θ I t ) ϑ ( − θ I t ) + (cid:89) I =1 ϑ ( − θ I t ) ϑ ( − θ I t ) − (cid:89) I =1 ϑ ( − θ I t ) ϑ ( − θ I t ) − (cid:35) . (3.26)Since the twists satisfy the condition (2.18), the Jacobi-Riemann identity (2.37), together with thefollowing parity properties ϑ ( − z ) = − ϑ ( z ) , ϑ ( − z ) = + ϑ ( z ) , ϑ ( − z ) = + ϑ ( z ) , ϑ ( − z ) = + ϑ ( z ) , (3.27)implies that Z ( − / = 0 . (3.28)The partition function for the other orientation is obtained from (3.24) by twisting the ϑ -functionswith θ I t and replacing M with N , namely Z / ( − = kN (cid:90) ∞ dt t (cid:34) (cid:89) I =1 ϑ ( θ I t ) ϑ ( θ I t ) + (cid:89) I =1 ϑ ( θ I t ) ϑ ( θ I t ) − (cid:89) I =1 ϑ ( θ I t ) ϑ ( θ I t ) + 1 (cid:35) . (3.29)Using the Jacobi-Riemann identity, we easily find Z / ( − = kN (cid:90) ∞ dt t . (3.30)Summing all contributions (3.20), (3.23), (3.25), (3.28) and (3.30), we conclude that the totalpartition function of the open strings with at least one end-point on the D-instantons is Z tot = k ( N − M ) (cid:90) ∞ dt t . (3.31)If we set N = M , (3.32)the total partition function vanishes and no forces are present. This no-force condition is equivalentto requiring that the brane system be conformal invariant. In the T-dual set-up, namely in the D0/D8-brane system, the vanishing of the partition function can be inter-preted as due to the creation of a fundamental string, see for example [38, 40, 42]. The instanton moduli and their action
In the previous sections we have described a system made of N D7-branes with flux, M D7 (cid:48) -branes without flux and k D-instantons. The flux on the D7-branes is subject to the condition(2.18) with all θ I different from zero in order to preserve the minimal amount of supersymmetry.For simplicity, from now on we choose the twists to be all equal, namely θ = θ = θ = θ = 14 . (4.1)With this choice the four complex directions are treated symmetrically and many formulas simplify.For example, the fermionic twist fields to be used in the vertex operators of the (–1)/7 strings (see(3.11) and (3.12)) becomeΣ( z ) = e i4 ( ϕ ( z )+ ϕ ( z )+ ϕ ( z )+ ϕ ( z )) and Σ( z ) = e − i4 ( ϕ ( z )+ ϕ ( z )+ ϕ ( z )+ ϕ ( z )) (4.2)in the NS and R sectors, respectively. The same twist fields are used also in the vertex operatorsof the 7/(–1) strings (see (3.14) and (3.15)), but in this case Σ( z ) is used in the R sector and Σ( z )in the NS sector. Both Σ( z ) and Σ( z ) are conformal fields of weight . It is also convenient todefine the bosonic twist fields∆( z ) = (cid:89) I =4 σ I ( z ) and ∆( z ) = (cid:89) I =4 σ I − ( z ) (4.3)to be used for the (–1)/7 and 7/(–1) strings respectively. These fields have conformal dimension and are conjugate to each other. Similarly, for the strings of type (–1)/7 (cid:48) and 7 (cid:48) /(–1) we definethe following pair of conjugate twist fields∆ (cid:48) ( z ) = (cid:89) I =4 σ I ( z ) and ∆ (cid:48) ( z ) = (cid:89) I =4 σ I − ( z ) , (4.4)which have conformal weight .Using these notations, we summarize for future reference all physical states of the open stringswith at least one end-point on the D-instantons and their vertex operators in Table 2. The action of the ( − / ( −
1) moduli can be obtained by dimensionally reducing the N = 1 superYang-Mills action from ten to zero dimensions and, in the notations previously introduced, it readsas follows S ( − = 1 g tr (cid:26) (cid:2) B I , B I (cid:3)(cid:2) B J , B J (cid:3) − (cid:2) B I , B J (cid:3)(cid:2) B I , B J (cid:3) − (cid:2) ξ, B I (cid:3)(cid:2) χ, B I (cid:3) − (cid:2) χ, B I (cid:3)(cid:2) ξ, B I (cid:3) + 12 (cid:2) χ, ξ (cid:3) + 12 λ (cid:2) χ, λ (cid:3) + 12 η (cid:2) χ, η (cid:3) + 14 λ IJ (cid:2) χ, λ KL (cid:3) (cid:15) IJKL − M I (cid:2) ξ, M I (cid:3) + λ IJ (cid:16)(cid:2) B I , M J (cid:3) − (cid:2) B J , M I (cid:3) + (cid:15) IJKL (cid:2) B K , M L (cid:3)(cid:17) + ( η + i λ ) (cid:2) B I , M I (cid:3) + ( η − i λ ) (cid:2) B I , M I (cid:3)(cid:27) (4.5)16ector Moduli Vertex Operators Statistics Dimensions( − / ( − B I , B I B I √ Ψ I e − ϕ , B I √ Ψ I e − ϕ bosonic (length) − ξ , χ ξ Ψ e − ϕ , χ Ψ e − ϕ bosonic (length) − M I , M I M I S I S + e − ϕ , S I S + e − ϕ M I fermionic (length) − λ IJ λ IJ √ S IJ S − e − ϕ fermionic (length) − λ λ S − S √ S − e − ϕ fermionic (length) − η η S + S √ S − e − ϕ fermionic (length) − ( − / w w √ ∆ Σ e − ϕ bosonic (length) − µ µ ∆ Σ S + e − ϕ fermionic (length) − / ( − w w √ ∆ Σ e − ϕ bosonic (length) − µ µ ∆ Σ S + e − ϕ fermionic (length) − ( − / (cid:48) µ (cid:48) µ (cid:48) ∆ (cid:48) S − e − ϕ fermionic (length) − (cid:48) / ( − µ (cid:48) µ (cid:48) ∆ (cid:48) S − e − ϕ fermionic (length) − Table 2:
The physical moduli corresponding to open strings with at least one end-point on the D-instantons,and their vertex operators in the canonical superghost pictures, ( −
1) in the NS sector and ( − ) in the Rsector. The last two columns contain their statistics and their scaling dimensions. where g = g s π α (cid:48) (4.6)is the Yang-Mills coupling constant in zero dimensions.The action involving the moduli of the mixed sectors is S mixed = 1 g tr (cid:8) w ( χ ξ + ξ χ ) w − µ ξ µ − µ ( η + i λ ) w − w ( η − i λ ) µ + µ (cid:48) χ µ (cid:48) (cid:9) . (4.7)Using standard conformal field theory methods, one can verify that all cubic couplings in theabove actions can be obtained by computing the 3-point amplitudes among the vertex operatorsintroduced in the previous section. If one applies this method to compute the 4-point amplitudesand obtain the quartic couplings, one encounters divergent integrals over the vertex insertionpoints, as it has also been recently pointed out in [43]. To circumvent this problem, we adoptthe strategy already used in [6, 31, 34] and introduce auxiliary fields to disentangle the quarticinteractions. For our purposes it is enough to consider the first two quartic terms in (4.5). Theminimal set of auxiliary fields which are needed to decouple these interactions comprises seven17uxiliary fields, denoted by D and D IJ with D JI = − D IJ ( I, J = 1 , . . . , D is realwhile the six D IJ ’s are complex with their complex conjugates D IJ given by D IJ = 12 (cid:15) IJKL D KL . (4.8)Thus D IJ correspond to six real degrees of freedom. Using these auxiliary fields, to which weassign canonical scaling dimensions of (length) − , the first two terms of (4.5) can be replaced by1 g tr (cid:26) D IJ D KL (cid:15) IJKL − D IJ (cid:16)(cid:2) B I , B J (cid:3) + 12 (cid:15) IJKL (cid:2) B K , B L (cid:3)(cid:17) + 12 D − i D (cid:2) B I , B I (cid:3)(cid:27) . (4.9)Indeed, eliminating the auxiliary fields through their equations of motion and using the Jacobiidentity, one can show that (4.9) reduces exactly to the first two terms of (4.5).We observe that the cubic terms in (4.9) can be obtained by computing the couplings among thevertex operators V B I and V B I , given in (3.1), and the following vertex operators for the auxiliaryfields V D IJ = − D IJ (cid:16) Ψ I ( z )Ψ J ( z ) + 12 (cid:15) IJKL Ψ K ( z )Ψ L ( z ) (cid:17) ,V D = − i D I ( z )Ψ I ( z ) . (4.10)These vertices are in the (0)-superghost picture, as appropriate for auxiliary fields, and are con-formal operators of weight 1.The vertex V D has a non-vanishing coupling also with the vertex operators V w and V w of themixed bosonic moduli w and w ; this coupling leads to the following term1 g tr (cid:8) − i D w w (cid:9) (4.11)that has to be added to the moduli action (4.7). Furthermore, we find convenient to add also theterm 1 g tr (cid:8) h (cid:48) h (cid:48) (cid:9) (4.12)where h (cid:48) and its conjugate h (cid:48) are auxiliary fields with scaling dimensions of (length) − . Even ifthese fields look trivial since they do not interact with any other moduli, it is useful to introducethem for reasons that will be clear in a moment. Also these auxiliary fields can be described byvertex operators in the (0)-superghost picture, which are V h (cid:48) = h (cid:48) ∆ (cid:48) ( z ) S ( z ) + S ( z ) √ V h (cid:48) = h (cid:48) ∆ (cid:48) ( z ) S ( z ) + S ( z ) √ . (4.13)These operators are conformal fields of weight 1.We now introduce ADHM-like variables by means of the following rescalings B I → g B I , M I → g M I , w → g w , µ → g µ , µ (cid:48) → g µ (cid:48) , h (cid:48) → g h (cid:48) (4.14)with analogous ones for their conjugates, in such a way that the rescaled bosons B I and w havedimensions of length, the rescaled fermions µ and µ (cid:48) have dimensions of (length) , and the rescaled18uxiliary field h (cid:48) becomes dimensionless. After these rescalings, adding all contributions one findsthat the instanton moduli action can be written as S inst = 1 g S G + S K + S D (4.15)where S G = tr (cid:26) D IJ D KL (cid:15) IJKL + 12 D + 12 (cid:2) χ, ξ (cid:3) + 12 λ (cid:2) χ, λ (cid:3) + 12 η (cid:2) χ, η (cid:3) + 14 λ IJ (cid:2) χ, λ KL (cid:3) (cid:15) IJKL (cid:27) , (4.16a) S K = tr (cid:26) − (cid:2) ξ, B I (cid:3)(cid:2) χ, B I (cid:3) − (cid:2) χ, B I (cid:3)(cid:2) ξ, B I (cid:3) − M I (cid:2) ξ, M I (cid:3) − µ ξ µ + w ξ χ w + w χ ξ w + µ (cid:48) χ µ (cid:48) + h (cid:48) h (cid:48) (cid:27) , (4.16b) S D = tr (cid:26) − i D (cid:16)(cid:2) B I , B I (cid:3) + w w (cid:17) − D IJ (cid:16)(cid:2) B I , B J (cid:3) + 12 (cid:15) IJKL (cid:2) B K , B L (cid:3)(cid:17) + λ IJ (cid:16)(cid:2) B I , M J (cid:3) − (cid:2) B J , M I (cid:3) + (cid:15) IJKL (cid:2) B K , M L (cid:3)(cid:17) + ( η + i λ ) (cid:16)(cid:2) B I , M I (cid:3) + w µ (cid:17) + ( η − i λ ) (cid:16)(cid:2) B I , M I (cid:3) − µ w (cid:17)(cid:27) . (4.16c)In the field theory limit α (cid:48) →
0, or equivalently in the strong-coupling limit g → ∞ , the term S G can be discarded and the fields D and D IJ become Lagrange multipliers for the bosonic constraints (cid:2) B I , B I (cid:3) + w w = 0 , (cid:2) B I , B J (cid:3) + 12 (cid:15) IJKL (cid:2) B K , B L (cid:3) = 0 , (4.17)while the fields ( η ± i λ ) and λ IJ become Lagrange multipliers for the fermionic constraints (cid:2) B I , M I (cid:3) + w µ = 0 , (cid:2) B I , M J (cid:3) − (cid:2) B J , M I (cid:3) + (cid:15) IJKL (cid:2) B K , M L (cid:3) = 0 . (4.18)In Section 6 we will show that the same equations arise from the ADHM construction for instantonsin eight dimensions. This fact allows us to regard the open strings ending on the D(–1) branes asinstanton moduli. ε -background We now generalize the action (4.15) by introducing the interactions with external backgrounds.From the string theory point of view, these backgrounds correspond either to vacuum expec-tation values of massless fields propagating on the world-volume of the D7 and D7 (cid:48) -branes, or tovacuum expectation values of bulk fields in the closed string sectors. Therefore, one can obtaintheir interactions with the instanton moduli by following the procedure described in [6, 16, 31],namely by computing the correlation functions among the vertex operators of the moduli and thevertex operators of the external background fields. However, there is also an alternative route toobtain these interactions which exploits the symmetries of the brane system.19et us give some details. The moduli actions (4.16) are invariant by construction under thesymmetry group U( k ) of the D-instantons, and the moduli χ and ξ can be interpreted as theparameters of infinitesimal U( k ) transformations . In our conventions, these transformations are T χ [ • ] = − [ χ, • ] if • is an adjoint modulus , − χ • if • is a fundamental modulus , • χ if • is an anti-fundamental modulus , (4.19)and similarly for T ξ . Using this notation, for example the terms − (cid:2) ξ, B I (cid:3)(cid:2) χ, B I (cid:3) and µ (cid:48) χ µ (cid:48) appearing in (4.16b) can be rewritten respectively as − T ξ [ B I ] T χ [ B I ] and − µ (cid:48) T χ [ µ (cid:48) ] . (4.20)All other χ - and ξ -dependent terms in (4.16) can be treated in the same way.The idea is to extend this approach to all symmetries of the brane system. The actions(4.16) are invariant under the U( N ) transformations of the N D7-branes and under the U( M )transformations of the M D7 (cid:48) -branes, in which only the mixed moduli transform according tothe representations defined by the open string construction. The actions (4.16) are also invariantunder the SU(4) transformations related to the rotations in the four complex planes indexed by I .Let us denote by ε I the parameters of such rotations and define the quantities ε = ε + ε + ε + ε , (4.21)and ε IJ = ε I + ε J − ε . (4.22)Then, it is easy to show that the actions (4.16) are invariant under the following transformations( B I , M I ) → e +i ε I ( B I , M I ) , ( B I , M I ) → e − i ε I ( B I , M I ) , ( λ IJ , D IJ ) → e +i ε IJ ( λ IJ , D IJ ) , (4.23)with all other moduli unchanged, provided ε = 0. This is a SU(4) symmetry whose origin isquite clear. Indeed, the initial SO(8) symmetry of the D-brane system is broken to SO(7) onthe D-instantons [16, 21], and this symmetry is further reduced to SO(6) (cid:39) SU(4) by the complexstructure we have introduced. Therefore, the four parameters ε I subject to the condition ε = 0can be understood as the parameters of the SU(4) transformations along the Cartan directions(see Appendix A).We now exploit the U( N ), U( M ) and SU(4) symmetries and replace the U( k ) transformations T χ and T ξ according to T χ [ • ] → T ( χ,a,m,ε I ) [ • ] , T ξ [ • ] → T ( ξ,a,m,ε I ) [ • ] (4.24) The use of χ or ξ is fixed by a neutrality condition with respect to the charge in the fifth complex plane of theinitial string construction. T ( χ,a,m,ε I ) and T ( ξ,a,m,ε I ) denote U( k ) × U( N ) × U( M ) × SU(4) infinitesimal transforma-tions parametrized by χ , a , m and ε I , and by ξ , a , m and ε I respectively, in the appropriaterepresentations. For example, applying this rule the terms in (4.20) get shifted and become − T ( ξ,a,m,ε I ) [ B I ] T ( χ,a,m,ε I ) [ B I ] = − (cid:0) − (cid:2) ξ, B I (cid:3) + ε I B I (cid:1)(cid:0) − (cid:2) χ, B I (cid:3) − ε I B I (cid:1) , − µ (cid:48) T ( χ,a,m,ε I ) [ µ (cid:48) ] = − µ (cid:48) ( − χ µ (cid:48) + µ (cid:48) m ) . (4.25)Proceeding systematically in this way, we can generate new terms in the moduli action. Takingfor simplicity the U( N ) and U( M ) parameters only along the Cartan directions, these new termscorrespond to shifting the actions (4.16) as follows S G → S (cid:48) G = S G + tr (cid:26) (cid:88) I 22o that the interpretation of χ as a parameter for the U( k ) transformations can be maintained.On the contrary, we find Q ξ = η . (4.37)The choice of the BRST charge Q introduces therefore an asymmetry between χ and ξ , which upto this point played a quite similar role.To obtain the BRST transformation of Y , we impose the nilpotency of Q up to symmetries,namely Q Y = Q X = T ( χ,a,m,ε I ) [ X ] . (4.38)Following this method, for the moduli of the ( − / ( − 1) sector we explicitly find Q B I = M I , Q M I = − (cid:2) χ, B I (cid:3) + ε I B I , Q B I = M I , Q M I = − (cid:2) χ, B I (cid:3) − ε I B I , Q λ IJ = D IJ , Q D IJ = − (cid:2) χ, λ IJ (cid:3) + ε IJ λ IJ , Q λ = D , Q D = − (cid:2) χ, λ (cid:3) , Q ξ = η , Q η = − (cid:2) χ, ξ (cid:3) . (4.39)with the understanding the repeated indices are not summed. For the mixed moduli, instead, weobtain Q w = µ , Q µ = − χ w + w a , Q w = µ , Q µ = w χ − a w , Q µ (cid:48) = h (cid:48) , Q h (cid:48) = − χ µ (cid:48) + µ (cid:48) m , Q µ (cid:48) = h (cid:48) , Q h (cid:48) = µ (cid:48) χ − m µ (cid:48) . (4.40)These BRST transformations allow us to build doublets of the form ( X, Q X ). They are( B I , M I ) , ( B I , M I ) , ( ξ, η ) , ( λ IJ , D IJ ) , ( λ, D ) (4.41)in the ( − / ( − 1) sector, and ( w, µ ) , ( w, µ ) , ( µ (cid:48) , h (cid:48) ) , ( µ (cid:48) , h (cid:48) ) (4.42)in the mixed sectors. Furthermore, we can check that the action (4.31) is BRST invariant and Q -exact. Indeed, Q S (cid:48) inst = 0 , S (cid:48) inst = Q Λ (4.43)where Λ = tr (cid:26) g (cid:16) λ IJ D KL (cid:15) IJKL + 12 λ D − η (cid:2) χ, ξ (cid:3) − i ε λ ξ (cid:17) + ξ (cid:16)(cid:2) B I , M I (cid:3) + (cid:2) B I , M I (cid:3) + w µ − µ w (cid:17) + a (cid:0) w µ − µ w (cid:1) − λ IJ (cid:16)(cid:2) B I , B J (cid:3) + 12 (cid:15) IJKL (cid:2) B K , B L (cid:3)(cid:17) − i λ (cid:16)(cid:2) B I , B I (cid:3) + w w (cid:17) − (cid:88) I ε I (cid:0) B I M I − M I B I (cid:1) + 12 µ (cid:48) h (cid:48) + 12 h (cid:48) µ (cid:48) (cid:27) . (4.44)We observe that the dependence of the moduli action on a , ε I and ε is entirely encoded in thefermion Λ on which the BRST charge acts. This fact will be important in the following, andin particular it suggests that the end-result of the localization process will not carry any explicit dependence on these parameters. 23 The instanton partition function The effects of the D-instantons in the D7/D7 (cid:48) system are encoded in the gran-canonical partitionfunction Z = ∞ (cid:88) k =0 q k Z k (5.1)where q = e π i τ is the instanton fugacity and Z k is the partition function in the k -instanton sector(with the convention that Z = 1). The latter is defined by the following integral Z k = (cid:90) d M k e − S (cid:48) inst . (5.2)where M k denotes the set of moduli with k instantons. Actually, as we have seen in the previoussection, all such moduli except for χ are paired in BRST doublets each one containing a bosonand a fermion. We have in fact the following structure d M k = dχ ( dB I dM I )( dB I dM I )( dλ IJ dD IJ )( dλ dD )( dξ dη )( dw dµ )( dw dµ )( dµ (cid:48) dh (cid:48) )( dµ (cid:48) dh (cid:48) ) . (5.3)Therefore, any rescaling of the doublets cancels between bosons and fermions and does not affectthe end result. We exploit this feature and rescale all doublets with a factor of 1 /x . In the limit x → ∞ a drastic simplification occurs since only the quadratic terms of the action S (cid:48) inst are relevantin the limit, while the cubic and quartic terms become subleading and can be neglected. In otherwords the moduli action becomes S (cid:48) inst = 1 x g tr (cid:26) D IJ D KL (cid:15) IJKL + 12 D + 12 (cid:2) χ, ξ (cid:3) + 12 λ (cid:2) χ, λ (cid:3) + 12 η (cid:2) χ, η (cid:3) + 14 λ IJ (cid:2) χ, λ KL (cid:3) (cid:15) IJKL + (cid:88) I 2. The first pole location corresponds to a pair of tableaux with one box each; theother locations describe pairs of tableaux, one with two boxes and the other with no boxes. Thisexample can be systematically generalized to other values of N and k .The evaluation of the residues is straightforward, even if tedious. When we sum up all contri-butions, remarkable simplifications occur and, as anticipated, the dependence on a u , m u and ε I inthe final result is only through the combination VM . We have explicitly checked this fact up to k = 5, and found Z = 12 ( VM ) + 52 VM ,Z = − 16 ( VM ) − 52 ( VM ) − VM ,Z = 124 ( VM ) + 54 ( VM ) + 15524 ( VM ) + 214 VM ,Z = − VM ) − 512 ( VM ) − VM ) − VM ) − VM . (5.24) This is fully equivalent to the so-called Jeffrey-Kirwan prescription [45]. Z k for the rank 1 theory arein one-to-one correspondence with the solid partitions of k . However, it is the generating functionof the planar partitions M ( q ) that appears in the end result (5.14). To understand this fact, letus specialize to N = 1 with a = 0 and m = ε , and then take the limit ε → 0. This limit clearlyimplies that VM → 1. It turns out that in this scaling all contributions to Z k corresponding toYoung tableaux which contain at least one box with n > n = 1 are uniformly weighted with ( − k . In otherwords, in this limit the gran-canonical partition function (5.1) can be written as a weighted sumover the solid partitions π : Z = (cid:88) π w π q | π | (5.25)where | π | is the size of π , and w π = (cid:40) ( − | π | if Y ( π ) does not extend in the ε direction , Y ( π ) does extend in the ε direction , (5.26)with Y ( π ) denoting the four-dimensional Young diagram associated to π . Given the form of theweight function (5.26), it is clear that only the solid partitions π which are also planar with respectto the ε -direction, contribute in the sum. Therefore, (5.25) actually coincides with the partitionfunction of the planar partitions with uniform weight (up to an alternating sign), which are countedby the MacMahon function, i.e. Z = M ( − q ) (5.27)in agreement with (5.14) when VM = 1. This argument can be extended to more general casesalong the lines discussed in [28]. We have anticipated that the physical states of the open strings with at least one end-point on aD(–1)-brane can be put in correspondence with the moduli of instantonic configurations in eightdimensions. In this section we would like to elaborate on this.There are different definitions of eight-dimensional gauge instantons, which are briefly reviewedin a similar (although not identical) brane context in [34]. The definition appropriate to our currentset-up is the one in which these configurations correspond to Yang Mills connections whose fieldstrength satisfies the (anti) self-duality condition ∗ ( F ∧ F ) = ± F ∧ F . (6.1)When this condition holds, the quartic action S for the gauge fields on the D7 branes, given in(2.4), drastically simplifies. Indeed, in this casetr (cid:0) t F (cid:1) = ± 12 tr( F ∧ F ∧ F ∧ F ) , (6.2)28nd hence S = (cid:40) − π i τ k , − π i τ k , (6.3)where τ is defined in (2.10) and k = 14!(2 π ) (cid:90) tr ( F ∧ F ∧ F ∧ F ) (6.4)is the fourth Chern number. This precisely matches the action of k anti D-instantons or k D-instantons, respectively.In analogy with the familiar case of the four-dimensional instanton, one way to realize theconditions (6.1) is to exploit the duality properties of the eight-dimensional Dirac matrices. Indeed,the chiral blocks σ µν (see (A.16)), satisfy σ [ µ µ σ µ µ ] = + 14! (cid:15) µ µ ...µ µ σ µ µ σ µ µ (6.5)where in the left-hand side the indices are all anti-symmetrized, while the anti-chiral blocks σ µν obey σ [ µ µ σ µ µ ] = − (cid:15) µ µ ...µ µ σ µ µ σ µ µ . (6.6)A field strength F µν proportional to σ µν or to σ µν would thus enjoy the property (6.1).To obtain such a field strength we can follow the ADHM construction in four dimensions (fordefiniteness we choose the minus sign in (6.1)). We first introduce the ADHM matrix ∆ = (cid:32) α [ N × k ] ( B − x ) [8 k × k ] (cid:33) (6.7)where α is a N × k matrix, while B and x are 8 k × k matrices given by B = ( B µ ) [ k × k ] ⊗ σ µ , x = x µ [ k × k ] ⊗ σ µ . (6.8)The matrices α and B µ contain the bosonic moduli of the configuration, while x µ are the coordi-nates in eight dimensions. Then, we define the gauge connection A µ , expressed as a N × N matrix,according to A µ = U † ∂ µ U , (6.9)where U = (cid:18) u [ N × N ] v [8 k × N ] (cid:19) (6.10)is a ( N + 8 k ) × N matrix subject to the following conditions∆ † U = U † ∆ = 0 , U † U = [ N × N ] . (6.11) Since the construction involves matrices of various sizes, sometimes we find useful to indicate explicitly theirsizes with the notation A [ n × n ] for a matrix A of size n × n . F µν = ∂ µ A ν − ∂ ν A µ + (cid:2) A µ , A ν (cid:3) = 2 ∂ [ µ U † ∂ ν ] U + 2 U † ∂ [ µ U U † ∂ ν ] U = 2 ∂ [ µ U † (cid:0) − U U † (cid:1) ∂ ν ] U . (6.12)We now introduce a 8 k × k matrix M such that − U U † = ∆ M ∆ † (6.13)or equivalently, thanks to the conditions (6.11), such that∆ † ∆ = M − . (6.14)Plugging (6.13) into (6.12) leads to F µν = 2 ∂ [ µ U † ∆ M ∆ † ∂ ν ] U = 2 U † ∂ [ µ ∆ M ∂ ν ] ∆ † U = 2 U † (cid:18) σ [ µ M σ ν ] (cid:19) U = 2 v σ [ µ M σ ν ] v (6.15)where in the second line, with an abuse of notation, we have denoted [ k × k ] ⊗ ( σ µ ) [8 × and [ k × k ] ⊗ ( σ µ ) [8 × simply as σ µ and σ ν .If N ≥ 8, it is possible to require that the N × N matrix 2 v σ [ µ M σ ν ] v has a single non-trivialblock of size (8 × 8) proportional to σ µν . If this is the case, then, the field strength (6.15) satisfiesthe relation (6.1) with the minus sign as a consequence of the anti-self duality property (6.6). Ofcourse, this requirement puts stringent constraints on the form of the matrices M and ∆. For k = 1, a solution is obtained by taking M = f [8 × , α = (cid:18) ρ [8 × [( N − × (cid:19) , v = f α ,u = f (cid:18) − ( B µ − x µ ) σ µ [8 × ( N − [( N − × [( N − × ( N − (cid:19) (6.16)with f = ( ρ + r ) − and r = ( x µ − B µ )( x µ − B µ ). Through (6.15), this leads to the followingfield strength F µν = 2 ρ ( ρ + r ) (cid:18) σ µν [8 × ( N − [( N − × [( N − × ( N − (cid:19) . (6.17)This represents an instanton solution of size ρ and center B µ which corresponds to the embeddingof the SO(8) octonionic field strength of [46, 47] into U( N ) (with N ≥ A µ = ( x ν − B ν ) ρ + r (cid:18) σ νµ [8 × ( N − [( N − × [( N − × ( N − (cid:19) , (6.18)and that the fourth Chern number (6.4) is one [46, 47]. Generalizations to higher k are possible.In a supersymmetric theory, like the one we are considering, instantons preserve a fraction ofsupersymmetry. A supersymmetric instanton is then characterized by the existence of a Killingspinor e such that the supersymmetry variation of the gaugino Λ vanishes, namely δ Λ = 12 F µν γ µν e = 0 , (6.19)30hich reduces to F µν σ µν e = 0 (6.20)for a chiral e . In [32] it has been shown that to solve this equation it is enough to require that∆ † ∆ e = f − k × k ] ⊗ e (6.21)for some ( k × k ) matrix f . This means that the matrix M introduced in (6.14) must be such that M e = f [ k × k ] ⊗ e .To make contact with the previous sections, we write the matrices B and α appearing in theADHM matrix ∆ in terms of the bosonic moduli corresponding to the open strings in the D(–1)/D7brane system. In particular we take the eight k × k matrices B µ , used to construct B as in (6.8), tobe given by the neutral moduli B I and B I of the D(–1)/D(–1) sector, and take the N × k matrix α to be related to the mixed moduli w of the D7/D(–1) sector according to α = w [ N × k ] ⊗ ψ † [1 × where ψ a reference Weyl chiral spinor [32]. Given the choices we made in the string construction,we take ψ = e + i e (cid:48) (6.22)with e = δ a, and e (cid:48) = δ a, . The choice on the Killing spinor e reflects our choice of the preservedsupersymmetry (and hence of the BRST charge) which, in turn, is related to how SO(8) has beenbroken to SO(7). The second choice on e (cid:48) reflects the choice of the complex structure with whichwe have broken SO(7) to SU(4) (see Appendix A). With these positions it is easy to verify that ψ ψ † e = − τ e + e and σ µν e = − ( τ m ) µν τ m e (6.23)where τ m are the seven octonionic matrices defined in (A.13). Furthermore, we have∆ † ∆ = α † α + ( B † − x † )( B − x )= ww ⊗ ψψ † + 12 (cid:2) B µ , B ν ] ⊗ σ µν + ( B µ − x µ )( B µ − x µ ) ⊗ [8 × , (6.24)and thus projecting onto e and using the relations (6.23), we deduce that in order to satisfy (6.21)one must require that 12 (cid:2) B µ , B ν ]( τ m ) µν + ww δ m = 0 . (6.25)In complex notation, these equations read (cid:2) B I , B I (cid:3) + w w = 0 , (cid:2) B I , B J (cid:3) + 12 (cid:15) IJKL (cid:2) B K , B L (cid:3) = 0 , (6.26)which are exactly the constraints reported in (4.17) and derived from the moduli action in theD(–1)/D7-brane system.We observe that for k = 1 the first constraint leads to w = w = 0. This implies that the ADHMdata reduce to those in (6.16) for ρ = 0, thus describing a point-like instanton configurationwith vanishing size. It would be interesting to explore what happens if one introduces a non-commutative deformation parameter in such a way that the right-hand side of the real constraintin (6.26) differs from zero, thus allowing to have configurations with non-zero size, and moregenerally to investigate the features of the solutions of the ADHM constraints in the general case.We leave these issues to future investigations. 31 Conclusions In this paper we have studied a D(–1)/D7-brane system in Type IIB string theory that describesthe non-perturbative sector of a U( N ) gauge theory in eight dimensions, and provided an explicitanalysis of the massless open string states and of their vertex operators in the various sectors. Thenew ingredient with respect to previous work is the introduction of a constant magnetic flux onthe world-volume of the D7-branes which allows for the existence of bosonic moduli in the mixedsectors. After discussing the moduli action, we have computed the instanton partition functionusing localization and confirmed the results of [27] obtaining a closed form expression in terms ofthe MacMahon function.It is important to remark that this instanton partition function and the corresponding prepo-tential given in (5.16) involve only the fields of the overall U(1) factor inside U( N ). Furthermore,the prepotential is cubic in the ε I parameters used in the localization process, showing that thegauge effective action receives non-perturbative corrections only in curved space. To see this inan explicit way, let us first turn off the parameters m u so that the quantity M in (5.13) simplyreduces to tr φ , where φ is the scalar component of the vector superfield (see (2.8)). Then let usnote that ( ε + ε )( ε + ε )( ε + ε ) ∼ t µ ...µ W µ µ W µ µ W µ µ I µ µ (7.1)where W µν ≡ W µνz is the graviphoton field strength defined in (4.27a). The tensor I µν , instead,describes the background magnetic flux on the D7-branes and is obtained from (2.12) by setting f I = − I = 1 , . . . . Finally, t is the tensor that typically appears in higher derivativeactions, like for instance (2.4) . Suppressing all indices for simplicity, the prepotential (5.16) istherefore F ∼ t W I tr φ log M ( − q ) . (7.2)To obtain the effective action, we promote φ to the full scalar superfield Φ( x, θ ) given in (2.8) andthe graviphoton field strength W µν to the full graviphoton superfield [31] W µν ( x, θ ) = W µν ( x ) + θ χ µν ( x ) + 12 θσ λρ θ R µνλρ ( x ) + . . . (7.3)where χ µν ( x ) and R µνρσ ( x ) are respectively, the self-dual parts of the gravitino field strength andof the Riemann tensor. In this way we obtain a prepotential F (Φ , W ) that is linear in Φ and cubicin W . The eight-dimensional effective action is then given by (cid:90) d x d θ F (Φ , W ) + c . c . (7.4)After integration over the θ ’s, our results predict an all-instanton formula for higher derivativecouplings in Type II B supergravity, which are schematically of the form t t R tr F F class , inpresence of D7-branes carrying a non-trivial world-volume flux F class ∼ I along a U(1) subgroupof U( N ). It would be very interesting to further investigate the properties and the implications of According to (2.14), this corresponds to the choice of the twist parameters θ I = 1 / Given the particular form of W µν and I µν , one has t µ ...µ W µ µ W µ µ W µ µ I µ µ ∼ (cid:0) W W W I (cid:1) − 12 tr (cid:0) W W (cid:1) tr (cid:0) W I (cid:1) where the trace is over the space-time indices, from which the result (7.1) easily follows. ε I do not add up to zero. Acknowledgments We would like to thank Igor Pesando for helpful discussions. This research is partly supportedby the INFN Iniziativa Specifica ST&FI “String Theory & Fundamental Interactions”. A Notations and conventions In this appendix we collect our notations for the various symmetry groups of the brane systemand for the spinors and Dirac matrices. From ten to eight dimensions The symmetry of the ten-dimensional flat background in which the superstring theory is definedis broken by the presence of the seven-branes asSO(10) → SO(8) ⊗ SO(2) , (A.1)with SO(8) rotating the first eight coordinates x µ . Thus, the vector representation decomposes as → ( v , ⊕ ( , +1) ⊕ ( , − , (A.2)where each couple ( R , p ) indicates a representation R of SO(8) and a charge p under U(1) ∼ SO(2).The U(1) Cartan subgroup of SO(8) corresponds to the rotations in the ( x I − , x I ) planes, with I = 1 , . . . 4, and here we will denote its four parameters as ε I .Under the breaking (A.1), the 32-components of the ten dimensional spinors (which can bechiral or anti-chiral) decompose into S A S ± , where S A are the 16 components of SO(8) spinors(chiral or anti-chiral) and S ± have charge ± / s → (cid:0) s , +1 / (cid:1) ⊕ (cid:0) c , − / (cid:1) , c → (cid:0) s , − / (cid:1) ⊕ (cid:0) c , +1 / (cid:1) . (A.3)The ten-dimensional Dirac matrices take the formΓ µ = γ µ ⊗ , Γ = γ ⊗ σ , Γ = γ ⊗ σ , (A.4)where γ µ with µ = 1 , . . . γ is the eight-dimensionalchirality matrix and the sigmas are the usual Pauli matrices. The ten-dimensional charge conju-gation matrix C satisfying C Γ M (cid:0) C − (cid:1) T = − (cid:0) Γ M (cid:1) T for M = 1 , . . . , 10 is written as C = C ⊗ σ , (A.5)where C is the eight-dimensional charge conjugation which is such that C γ µ (cid:0) C − (cid:1) T = − ( γ µ ) T .33 ight-dimensional spinors For the eight-dimensional spinors we can use either the “spin field” basis or the Majorana-Weylbasis, which we are going to describe. • The spin field basis: In the “spin field” basis the spinor components are labeled by (twice)their SO(8) weight vectors and are S A = (cid:16) S ++++ , S − +++ , S + − ++ , S −− ++ , S ++ − + , S − + − + , S + −− + , S −−− + ,S +++ − , S − ++ − , S + − + − , S −− + − , S ++ −− , S − + −− , S + −−− , S −−−− (cid:17) . (A.6)In this basis the Dirac matrices are γ = σ ⊗ ⊗ ⊗ , γ = σ ⊗ ⊗ ⊗ ,γ = σ ⊗ σ ⊗ ⊗ , γ = σ ⊗ σ ⊗ ⊗ ,γ = σ ⊗ σ ⊗ σ ⊗ , γ = σ ⊗ σ ⊗ σ ⊗ ,γ = σ ⊗ σ ⊗ σ ⊗ σ , γ = σ ⊗ σ ⊗ σ ⊗ σ . (A.7)The chirality matrix is γ = γ γ γ γ γ γ γ γ = σ ⊗ σ ⊗ σ ⊗ σ , (A.8)while the charge conjugation matrix reads C = σ ⊗ σ ⊗ σ ⊗ σ . (A.9)The chirality matrix is diagonal, and the eight components in (A.6) with an even number ofminuses are chiral, i.e. they span the s representation of SO(8), while the eight ones with an oddnumber of minuses are anti-chiral, i.e. they span the c representation. • The Majorana-Weyl basis: For SO(8) it is also possible to use a Majorana-Weyl basis. Thiscan be achieved introducing the chiral combinations S α = 1 √ i S + −− + − i S − ++ − − S + −− + − S − ++ − − i S + − + − − i S − + − + − S + − + − + S − + − + − i S ++ −− + i S −− ++ − S ++ −− − S −− ++ − i S ++++ − i S −−−− S ++++ − S −−−− , (A.10)34nd the anti-chiral ones S ˙ α = 1 √ i S + −−− − i S − +++ − S + −−− − S − +++ − i S + − ++ − i S − + −− − S + − ++ + S − + −− − i S ++ − + + i S −− + − − S ++ − + − S −− + − − i S +++ − − i S −−− + S +++ − − S −−− + . (A.11)In the basis ( S α , S ˙ α ), the chirality matrix is ⊗ σ , the charge conjugation is the identity and allDirac matrices are purely imaginary. They take the form γ m = τ m ⊗ σ , γ = ⊗ σ , (A.12)where τ m , with m = 1 , . . . , 7, are ( − i times) seven-dimensional Dirac matrices. These latter haveelements ( τ m ) αβ = − i (cid:0) δ mα δ β − δ mβ δ α + c mαβ (cid:1) , (A.13)where the completely anti-symmetric tensor c mnp describes the octonionic structure constants: c = c = c = c = c = c = c = 1 , (A.14)with all other elements being zero. The Dirac matrices in (A.12) can also be written as γ µ = (cid:18) σ µ σ µ (cid:19) , (A.15)with σ µ = (cid:0) τ m , − i (cid:1) and σ µ = (cid:0) τ m , i (cid:1) . We also write γ µν = 12 (cid:2) γ µ , γ ν (cid:3) = (cid:18) σ µν σ µν (cid:19) , (A.16)with σ mn = σ mn = τ mn = 12 (cid:2) τ m , τ n (cid:3) , σ m = − σ m = i τ m . (A.17)This Majorana-Weyl basis is the one used in [16]. Using (A.10) and (A.11) we can rewrite allquantities involving fermions given there, in the spin field basis used in this paper. With an abuse of notation we denote these matrices again by γ µ , even if they do not coincide with the ones in(A.7); rather they are equivalent to the latter under the change of basis in (A.10) and (A.11). he U(4) symmetry The SO(8) symmetry contains a U(4) subgroup that preserves the complex structure introducedin (2.13). The U(1) Cartan subgroup of SO(8) in in U(4) and acts on the complex coordinates as z I → e i ε I z I . (A.18)One has U(4) (cid:39) (cid:0) SU(4) ⊗ U(1) (cid:1) / Z , where U (1) is the diagonal action in the U(1) subgroupparametrized by ε/ 4, where ε = ε + ε + ε + ε . (A.19)The Cartan subgroup of SU(4) is spanned by the “traceless” parameters (cid:98) ε I = ε I − ε/ 4, out of whichonly three are independent. We can denote the U(4) representations by r q , where r is a SU(4)representation and q a U(1) charge. With respect to U(4), the relevant SO(8) representationsdecompose according to the following pattern. For the vector we have v → ⊕ − , (A.20)the two representations in the right-hand side corresponding to z I and ¯ z I . For the chiral spinorwe have s → ⊕ ⊕ − . (A.21)The two SU(4) singlets are, respectively, S = S ++++ , S = − S −−−− , (A.22)while the states of the representation are S IJ with I, J = 1 , . . . S IJ = − S JI . Theiridentification in the spin field basis is S = S ++ −− , S = − S + − + − , S = S + −− + ,S = S − ++ − , S = − S − + − + , S = S −− ++ . (A.23)Thus S IJ is proportional to the spin field component which has a plus in the I -th and J -thdirections. The conjugates S IJ , which are the spin fields with a minus in the I -th and J -thdirections, are not independent. Indeed we have S IJ = 12 (cid:15) IJKL S KL . (A.24)The anti-chiral spinor representation decomposes as c → ⊕ − . (A.25) We fix the convention on the U(1) charge by declaring that the charge of the fundamental representation, whosestates rotate by exp(i ε/ The minus signs appearing in (A.22) and later in (A.23) and (A.26) are inserted to be consistent with the actionof the charge conjugation matrix (A.9) on the spin fields. S I = S − +++ S + − ++ S ++ − + S +++ − , S I = S + −−− − S − + −− S −− + − − S −−− + . (A.26)Using first the change of basis (A.10) and (A.11), and then the identifications (A.22), (A.23) and(A.26) we can express all quantities involving the fermions given in [16] in terms of the U(4)notation used in the main text. The SO(7) symmetry The moduli action described in Section 4 possesses a BRST charge which in our conventions is thelast component S of a chiral spinor (see (4.32) and (4.33)). The SO(8) subgroup that preservesthe BRST charge is SO(7), embedded in a non-standard fashion in which the chiral spinordecomposes as s → ⊕ , where is the vector representation of SO(7). More explicitly, we have S α → ( S m , S ) with m = 1 , . . . , 7. Both the anti-chiral spinor c and the vector v becomespinors of SO(7). The adjoint representation of SO(8) decomposes according to → + ,where is the adjoint of SO(7). In particular, this means that an anti-symmetric tensor ofSO(8), like for example D µν , decomposes into D µν = 12 D mn ( τ mn ) µν , D µν = D m ( τ m ) µν . (A.27)Here D mn and D m are, respectively, an anti-symmetric tensor and a vector of SO(7), while τ m and τ mn are the matrices introduced in (A.13) and (A.17).This is another way in which the SO(7) symmetry emerges in our brane system and in theADHM construction described in section 6; indeed, to disentangle the quartic term proportionalto [ a µ , a ν ] [ a µ , a ν ] in the U( k ) moduli action, thanks to the Jacobi identity, the minimal requiredset of auxiliary fields is not a generic anti-symmetric tensor D µν but its part only. The U(1) Cartan subgroup of SO(7) is embedded into the Cartan subgroup of SO(8) by restricting the fourparameters (cid:15) I to be traceless, i.e. to satisfy the requirement ε = 0. Only in this case is thespinor S invariant. Note that this is the same condition that defines the Cartan subgroup of theSU(4) ⊂ U(4) symmetry. The SO(6) (cid:39) SU(4) symmetry The subgroup of SO(7) that is compatible with the complex structure (2.13) is the SO(6) groupunder which the SO(7) vector decomposes as → ⊕ , namely V m → ( V (cid:98) m , V ), with (cid:98) m = 1 , . . . → s ⊕ c , the two addends being thechiral and anti-chiral spinors of SO(6). The Cartan subgroup of this SO(6) coincides with the oneof SO(7). This embedding is exchanged by triality with the standard one in which v → ⊕ or with the embeddingwhere it is the anti-chiral spinor that decomposes in such a way. To be precise, given the vector v µ , it is ( v m , − v ) that transforms as a spinor. and ¯ , while a vector like D (cid:98) m is mapped to the anti-symmetric tensor D IJ , with I, J = 1 , . . . 4, satisfying the constraint (A.24). B Technical details In this appendix we collect some technical material related to the string theory and conformal fieldtheory methods used in the main text. Twisted coordinates and mode expansions In [35] the conformal field theory of bosonic and fermionic fields with non-trivial monodromyproperties has been described in full detail. Here we make use of the results of that reference andadapt them to the case of interest. The complex bosonic and fermionic string coordinates along thelongitudinal directions in the D7/D7 (cid:48) sectors satisfy the monodromy properties (2.15) and (2.16)around the origin of the world-sheet. After canonical quantization, these fields have the followingmode expansions i ∂Z I ( z ) = √ α (cid:48) (cid:16) ∞ (cid:88) n =1 a In − θ I z − n + θ I − + ∞ (cid:88) n =0 a † In + θ I z n + θ I − (cid:17) , i ∂Z I ( z ) = √ α (cid:48) (cid:16) ∞ (cid:88) n =0 a I, n + θ I z − n − θ I − + ∞ (cid:88) n =1 a † I, n − θ I z n − θ I − (cid:17) , (B.1)and Ψ I ( z ) = √ α (cid:48) ∞ (cid:88) n =0+ ν (cid:16) Ψ In − θ I z − n + θ I − + Ψ † In + θ I z n + θ I − (cid:17) , Ψ I ( z ) = √ α (cid:48) ∞ (cid:88) n =0+ ν (cid:16) Ψ I, n + θ I z − n − θ I − + ∞ (cid:88) n =0+ ν Ψ † I, n − θ I z n − θ I − (cid:17) (B.2)for the fermionic coordinates with ν = 0 in the R sector and ν = in the NS sector. The bosonicoscillators satisfy the canonical commutation relations (cid:104) a I, n + θ I , a † Jm + θ J (cid:105) = ( n + θ I ) δ JI δ nm ∀ n, m ≥ , (cid:104) a In − θ I , a † J, m − θ J (cid:105) = ( n − θ I ) δ IJ δ nm ∀ n, m ≥ , (B.3)while the fermionic oscillators obey the canonical anti-commutators (cid:110) Ψ I, n + θ I , Ψ † Jm + θ J (cid:111) = (cid:110) Ψ Jn − θ J , Ψ † I, m − θ I (cid:111) = δ JI δ nm ∀ n, m ≥ ν . (B.4)Notice that if all θ I ’s are different from zero (which is the case considered in the main text), thereare no bosonic oscillators with index 0, and hence the momentum can not be defined in any ofthese directions. 38ollowing [35], we define the Virasoro generators L n for the D7/D7 (cid:48) strings. In particular wefind that the bosonic contribution to L is L ( Z )0 = (cid:98) N ( Z ) + 12 (cid:88) I =1 θ I (1 − θ I ) (B.5)where (cid:98) N ( Z ) is the bosonic number operator (cid:98) N ( Z ) = (cid:88) I =1 (cid:20) ∞ (cid:88) n =1 a † I, n − θ I a In − θ I + ∞ (cid:88) n =0 a † In + θ I a I, n + θ I (cid:21) + ∞ (cid:88) n =1 (cid:0) a † , n a n + a † n a , n (cid:1) , (B.6)while the fermionic contribution to L is L (Ψ)0 = (cid:98) N (Ψ) + 12 (cid:88) I =1 θ I in the NS sector , (cid:98) N (Ψ) − (cid:88) I =1 θ I (1 − θ I ) in the R sector , (B.7)where the fermionic number operator (cid:98) N (Ψ) is (cid:98) N (Ψ) = (cid:88) I =1 (cid:20) ∞ (cid:88) n =1 − ν ( n − θ I )Ψ † I, n − θ I Ψ In − θ I + ∞ (cid:88) n =0+ ν ( n + θ I )Ψ † In + θ I Ψ I, n + θ I (cid:21) + ∞ (cid:88) n =0+ ν n (cid:0) Ψ † , n Ψ n + Ψ † n Ψ , n (cid:1) . (B.8)The c -numbers in (B.5) and (B.7) arise from the normal ordering of the bosonic and fermionicoscillators.The physical states must obey the conditions L ( Z )0 + L (Ψ)0 = 12 in the NS sector ,L ( Z )0 + L (Ψ)0 = 0 in the R sector , (B.9)which imply Eq. (2.19) of the main text. BRST variations We now give a few details on how the BRST transformations of the instanton moduli described inSection 4.3 can be derived using the vertex operators introduced in Section 3. To do so, we needthe OPE’s of the ten-dimensional string theory, like for example S ˙ A− / ( z ) S ˙ B− / ( w ) ∼ i ( C Γ M ) ˙ A ˙ B √ ψ M ( w ) e − ϕ ( w ) z − w + . . . (B.10)with M = 1 , . . . , 10. Here S ˙ A− / denotes a ten-dimensional spin field in the ( − )-superghostpicture with negative chirality. In particular, we have S ˙ A = S A S ± e − ϕ where S A is the eight-dimensional spin field defined in (A.6), with + or − depending on whether A is anti-chiral or39hiral. The Dirac matrices Γ M and the charge conjugation matrix C are given in (A.4) and (A.5).Using these matrices and the notation explained in Appendix A, from (B.10) we find for instance (cid:0) SS − e − ϕ (cid:1) ( z ) (cid:0) S I S + e − ϕ (cid:1) ( w ) ∼ − Ψ I ( w ) e − ϕ ( w ) z − w + . . . . (B.11)Let us now derive how the BRST charge (4.32) acts on the moduli B I and M I . As explainedin the main text, we need to compute (cid:104) Q , V M I ( w ) (cid:105) = 1 √ (cid:104) Q, V M I ( w ) (cid:105) + 1 √ (cid:104) Q, V M I ( w ) (cid:105) . (B.12)The second commutator vanishes and we are left with (cid:104) Q , V M I ( w ) (cid:105) = 1 √ (cid:104) Q, V M I ( w ) (cid:105) = − M I √ (cid:73) w dz π i (cid:0) SS − e − ϕ (cid:1) ( z ) (cid:0) S I S + e − ϕ (cid:1) ( w ) (B.13)where the minus sign is due to the fermionic statistics of M I and of the vertex operators. Usingthe OPE (B.11), we immediately find (cid:104) Q , V M I ( w ) (cid:105) = M I √ I ( w ) e − ϕ ( w ) . 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